In quantum optics, coherent energy exchanges may occur in the so called coherent regime of driving, where a strong, time-dependent field drives near resonant transitions in a quantum system, leading to the Rabi oscillations. In the additional presence of a thermal bath, the evolution of the quantum system is well described by the optical Bloch or Floquet master equations, depending on the strength of the driving. Such coherent energy exchanges play a central role in quantum technologies, which has motivated recent work on the consistency of the Bloch and Floquet master equations at the average level [1] and in the steady state [2].Here, we study coherent energy exchanges between a coherent field (laser) and an atom, at the fluctuating level. Studying the statistics of the work performed on the atom is particularly difficult when the laser is treated semiclassically: in this approach, the number of photons in the laser is assumed to be constant, but microscopically, the energy changes induced by the bath are associated to a variation of the number of photons. We overcome this difficulty by starting from a microscopic description of the laser. Using a two-point measurement technique with counting fields, we obtain the full counting statistics for the work performed by the coherent field on the atom. We derive a new fluctuation theorem, valid at all times, for quantum systems coupled to heat baths and work reservoirs. We also recover the Crooks relation, although the Hamiltonian of the total system is time independent.We then study the thermodynamic consistency of quantum optical master equations, and examine the thermodynamic consistency of the semiclassical treatment of the coherent field. The semiclassical approach was known to yield expressions of the work which seemed in contradiction with quantum thermodynamics predictions in the strong coupling regime [1]. We show that these apparent inconsistencies originate from the Born approximation, and show how to correctly derive the work. Finally, we find that the Floquet master equation is fully consistent, i.e., satisfies the first and second laws of thermodynamics at the fluctuating level, while the Bloch equation is only consistent at the average level.[1] C. Elouard, D. Herrera-Marti, M. Esposito, and A. Auffeves, “Thermodynamics of optical Bloch equations”, New Journal of Physics 22, 103039 (2020).[2] G. Bulnes Cuetara, A. Engel, and M. Esposito, “Stochastic thermodynamics of rapidly driven systems”, New J. Phys. 17, 055002 (2015)
